Failure of syndrome decoding

I am currently decoding a $$[8,4,4]$$ binary code using syndrome decoding, and I have two words I'd like to decode. Here is the parity-check matrix I am working with:

$$H = \begin{pmatrix} 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 \end{pmatrix}$$

For the first word $$y_1=(11010011)$$, the decoding algorithm didn't work because I didn't find the right syndrome. Indeed, $$Hy_1^T=(0110)$$ which is not in the syndrome table of $$H$$. For the second word $$y_2=(01010001)$$, the decoding worked and gave me a code element $$c=(11010001)$$. After looking at $$c$$, I realized that the first word I received was simply $$c-(00000010)$$, in other words there was a simple error of weight $$1$$. Is it normal that the algorithm couldn't decode it? Did I make an error computing the syndromes?

1 Answer

I get $$Hy_1^T = (0010)$$, I think you miscomputed.. Note that this is the 7th column of $$H$$ so the most likely error is $$(00000010)$$, which it indeed was.

• My bad, somehow three people were on this and we all got it wrong, wew. Thanks a lot! Jun 5, 2021 at 13:51